Do I own a domain if I register it?

Yes and no. Registering a domain name can be compared to a lease. When you register a domain you are paying for the right to use the domain for a specified period of time.  At the end of that time period you can choose to renew the domain and continue to use it or let the domain name expire.  Once a domain expires another person can register the domain and use it.

Since domains are not actually owned property, they can be disputed.  Disputing a domain registration is a challenging thing to do however it can be done.  Typically a domain dispute is the only way someone can take your domain away from you, unless you forget to renew it, and this typically only happens when registering a domain that has a trademark in it.   It is not suggested that you register a domain like pepsicola.com and expect to be able to keep it.